GLOBAL WELL-POSEDNESS FOR THE ... - Infoscience - EPFL
GLOBAL WELL-POSEDNESS FOR THE MAXWELL-KLEIN GORDON EQUATION IN 4 + 1 DIMENSIONS. SMALL ENERGY. JOACHIM KRIEGER, JACOB STERBENZ, AND DANIEL TATARU Abstract. We prove that the critical Maxwell-Klein Gordon equation on R4+1 is globally well-posed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.
1. Introduction Let R4+1 be the five dimensional Minkowski space equipped with the standard Lorentzian metric g = diag(1, −1, −1, −1, −1). Denote by φ : R4+1 → C a scalar function, and by Aα : R4+1 → R, α = 1 . . . , 4, a real valued connection form, interpreted as taking values in isu(1). Introducing the curvature tensor Fαβ := ∂α Aβ − ∂β Aα as well as the covariant derivative Dα φ := (∂α + iAα )φ the Maxwell-Klein Gordon system are the Euler-Lagrange equations associated with the formal Lagrangian action functional1 Z 1 1 1 L(Aα , φ) := Dα φDα φ + Fαβ F αβ dxdt; 2 R4+1 2 4 here we are using the standard convention for raising indices. Introducing the covariant wave operator 2A := Dα Dα we can write the Maxwell-Klein Gordon system in the following form (1)
∂ β Fαβ = −Jα := =(φDα φ), 2A φ = 0
One key feature of this system is the underlying Gauge invariance, which is manifested by the fact that if (Aα , φ) is a solution, then so is (Aα − ∂α χ, eiχ φ). This allows us to impose an The first author was partially supported by the Swiss National Science Foundation. The second author was partially supported by the NSF grant DMS-1001675 . The third author was partially supported by the NSF grant DMS-0801261 and by the Simons Foundation. The first author would like to express his thanks to the University of California, Berkeley, for hosting him during the summer 2011 and 2012. 1The Lagrangian below should also contain a mass term, which we choose to neglect here; thus the system under consideration might be more aptly named “the Maxwell-scalar field system”. For historical reasons we have chosen to retain the “Maxwell-Klein Gordon” terminology. 1
additional Gauge condition, and we shall henceforth impose the Coulomb Gauge condition which requires (2)
4 X
∂j Aj = 0
j=1
The MKG-CG system can be written explicitly in the following form (3a)
2Ai = Pi Jx
(3b)
2A φ = 0
(3c)
∆A0 = J0
(3d)
∆∂t A0 = ∇i Ji ,
where the operator P denotes the Leray projection onto divergence free vector fields, P = I − ∇∆−1 ∇ The last equation (3d) is a consequence of (3c) due to the divergence free condition on the moments ∂ α Jα = 0, which in turn follows from (3b). However, it plays a role in the sequel so it is added here for convenience. Here it is assumed that the data Aj (0, ·) satisfy the vanishing divergence relation (2). We remark that given an arbitrary finite energy data set for the MKG problem, one can find a gauge equivalent data set of comparable size which satisfies the Coulomb gauge condition. This argument only involves solving linear elliptic pde’s, and is omitted. The key question now is to decide whether a family of data (A, φ)[0] := (Aα (0, ·), ∂t Aα (0, ·), φ(0, ·), ∂t φ(0, ·)) which satisfy the compatibility conditions required by the two equations for A0 above can be extended to a global-in-time solution for the Maxwell-Klein-Gordon system. The key for deciding this question is the criticality character of the system. Note that the energy Z 1X 2 1X E(A, φ) := Fαβ + |Dα φ|2 dx 2 α R4 4 α,β is preserved under the flow (1), and in our 4 + 1-dimensional setting it is also invariant under the natural scaling φ(t, x) → λφ(λt, λx), Aα (t, x) → λA(λt, λx) This means the 4+1-MKG system is energy critical, and in recent years a general approach to the large data Cauchy problem associated with energy critical wave equations has emerged. The first key step in this approach consists in establishing an essentially optimal global wellposedness result for data which are small in the energy norm, which is usually the optimal small-data global well-posedness result achievable. In this paper, we set out to prove this for the 4 + 1-dimensional Maxwell-Klein-Gordon system. Theorem 1. a) Let (A, φ)[0] be a C ∞ Coulomb data set satisfying (4)
E(A, φ) < ∗
for a sufficiently small universal constant ∗ > 0. Then the system (1) admits a unique global smooth solution (A, φ) on R4+1 with these data. 2
b) In addition, the data to solution operator extends continuously2 on the set (4) to a map H˙ 1 (R4 ) × L2 (R4 ) 3 (A, φ)[0] → (A, φ) ∈ C(R; H˙ 1 (R4 )) ∩ C˙ 1 (R, L2 (R4 )) We remark that the same result holds in all higher dimensions for small data in the scale n n invariant space H˙ 2 −1 × H˙ 2 −2 . This has already been known in dimensions n ≥ 6, see [21]. We have chosen to restrict our exposition to the more difficult case n = 4 in order to keep the notations simple, but our analysis easily carries over to dimension n = 5. On the other hand, we do not know whether a similar result holds in dimension n = 3. Before explaining some more details of our approach, we recall here earlier developments on this problem, and how our approach relates to these. Considering the case of general spatial dimension n and denoting the critical Sobolev exponent by sc = n2 − 1 (thus sc = 1 for n = 4, corresponding to the energy), a global regularity result for data which are smooth and small in H˙ sc was established in dimensions n ≥ 6 in the work [21] which served as inspiration to the present work. We note in passing that a result analogous to [21] was established in [18] for the Yang-Mills system in dimensions n ≥ 6, and the present work most likely also admits a corresponding analogue for the 4 + 1-dimensional Yang-Mills problem. The global regularity question for the physically relevant n = 3 case of the Yang-Mills problem had been established earlier in the groundbreaking work [6]. Observe that this problem is energy sub-critical. In the context of MKG, the result [21] had been preceded by a number of works which aimed at improving the local wellposedness of MKG in the n = 3 case, beginning with [9], followed by [5], and more recently [19]; the latter in particular established an essentially optimal local well-posedness result by exploiting a subtle cancellation feature of MKG, which also plays a role in the present work. We also mention the recent result [8] which establishes global regularity for energy sub-critical data in the n = 3 case, in the spirit of earlier work by Bourgain [3]. In the higher dimensional case n ≥ 4, an essentially optimal local well-posedness result for a model problem closely related to MKG was obtained in [14]. This model problem does not display the crucial cancellation feature of the precise MKG-system which enable us here to go all the way to the critical exponent and global regularity. We mention also that essentially optimal local well-posedness for the exact MKG-system was obtained in [17]. Finally, the recent work [24] established global regularity of equations of MKG-type with data small in a weighted but scaling invariant Besov type space, in the case n = 4. The present paper follows a similar strategy as [21] : one observes that the spatial Gauge connection components Aj , j = 1, 2, 3, which are governed by the first equation (3a), may in fact be decomposed into a free wave part and an inhomogeneous term (the second term in the Duhamel formula for A) which in fact obeys a better l1 -Besov type bound (while energy corresponds to l2 ) Aj = Afree + Anonlin j j This is important for handling the key difficulty of the MKG-system , which is the equation for φ, i. e. the second equation (3b). In fact, we shall verify that the contribution of the term Anonlin to the difficult magnetic interaction term 2iAj ∂j φ in the low-high frequency interaction case can be suitably bounded when combined with the term 2iA0 ∂t φ, an observation coming from [19]. However, the contribution of the free term Afree to the magnetic interaction term 2here
the continuity is locally in time 3
is nonperturbative and cannot be handled in this manner. Thus, following the example set in [21], we retain this term into the covariant wave operator3 2Afree . More precisely, we shall define a suitable paradifferential wave operator 2pA which incorporates the ’leading part’ of 2Afree while relegating the rest to the source terms on the right. The key novelty of this paper then is the development of a functional calculus, involving in particular X s,b -type as well as atomic null-frame spaces developed in other contexts, for solutions of the general inhomogeneous ’covariant’ wave equation 2pA = f This refined functional calculus is necessary to control the nonlinear interaction terms, which become significantly more delicate in the critical dimension than in the setting studied in [21]. In particular, the Strichartz norms themselves appear far from sufficient to handle the present situation. The above covariant wave equation will be solved by means of a suitable approximate parametrix, and we show that this parametrix satisfies many of the same bounds as the usual free wave parametrix, in particular encompassing refined square sum type microlocalized Strichartz norms as well as null-frame spaces. We expect the calculus developed here to be of fundamental importance in other contexts, such as the regularity question of the critical Yang-Mills system and related problems from mathematical physics. 2. Technical preliminaries Throughout the sequel we shall rely on Littlewood-Paley calculus, both in space as well as space-time. In particular, we constantly invoke the standard Littlewood-Paley localizers Pk , k ∈ Z, which are defined by |ξ| ˆ d P k f = χ( k )f (ξ) 2 4 for functions f defined on R . Here χ is a smooth bump function, supported on [ 14 , 4], which P satisfies the key condition k∈Z χ( 2ξk ) = 1 if ξ > 0. To measure proximity of the space-time Fourier support to the light cone, we use the concept of modulation. Thus we introduce the multipliers Qj , j ∈ Z, via |τ | − |ξ| d Q )fˆ(τ, ξ) j f (τ, ξ) = χ( 2j with the same χ as before, whereˆin this context denotes the space-time Fourier transform. We then refer to 2j as the modulation of the function. On occasion we shall also use multipliers Sl , which restrict the space-time frequency to size ∼ 2l . These multipliers allow us to introduce a variety of norms. In particular, for any p ∈ [1, ∞), we set for any norm k · kS X
1
F p :=
Pk F p p l S
S
k∈Z
We also have the following X s,b -type norms, applied to functions localized to spatial frequency ∼ 2k : X
p 1 p
F kX s,r := 2sk 2rj Qj F 2 , p ∈ [1, ∞), Lt,x
p
j∈Z 3Here
we set Afree = 0. 0 4
with the obvious analogue
F kX s,r := 2sk sup 2rj Qj F 2 ∞ L
t,x
k∈Z
For more refined norms, we shall also have to use multipliers Plω , which localize the homoξ to caps ω ⊂ S 3 of diameter 2l , by means of smooth cutoffs. In these geneous variable |ξ| situations, we shall assume that for each l a uniformly (in l) finitely overlapping covering of S 3 by caps ω has been chosen with appropriate cutoffs subordinate to these caps. Similar comments apply to the multipliers PC l00 which localize to rectangular boxes and will be dek fined below. Given a norm k · kS with corresponding space S, we denote by Sk the space of functions in S which are localized to frequency ∼ 2k . Furthermore, we denote by Sk,± the subspace of functions in Sk with Fourier support in the half-space τ >< 0, with τ the Fourier variable dual to t. 3. Function Spaces There are three function spaces we work with: N , N ∗ , and S. These are set up to that their dyadic subspaces Nk , Nk∗ , and Sk satisfy the following relations: 0,− 21
(5)
0, 12
Nk = L1 (L2 ) + X1
,
k F k2N =
X
X1
⊆ Sk ⊆ Nk∗ ,
Then define: k Pk F k2Nk .
k
We also define
Sk]
by kukS ] = k2ukNk + k∇ukL∞ L2 k
On occasion we need to separate the two characteristic cones {τ = ±|ξ|}. Thus we define Nk = Nk,+ ∩ Nk,−
Nk,± , ] Sk,± , ∗ Nk,± ,
] ] Sk] = Sk,+ + Sk,− ∗ ∗ Nk∗ = Nk,+ + Nk,−
Our space Sk scales like L2 free waves, and is defined by: k φ k2Sk = k φ k2S str + k φ k2S ang + k φ k2 0, 12 , k
k
X∞
where: (6)
1
4
Skstr = ∩ 1 + 3/2 6 3 2( q + r −2)k Lq (Lr ) , q
r
k φ k2S ang = sup k
4
X
l
1. Introduction Let R4+1 be the five dimensional Minkowski space equipped with the standard Lorentzian metric g = diag(1, −1, −1, −1, −1). Denote by φ : R4+1 → C a scalar function, and by Aα : R4+1 → R, α = 1 . . . , 4, a real valued connection form, interpreted as taking values in isu(1). Introducing the curvature tensor Fαβ := ∂α Aβ − ∂β Aα as well as the covariant derivative Dα φ := (∂α + iAα )φ the Maxwell-Klein Gordon system are the Euler-Lagrange equations associated with the formal Lagrangian action functional1 Z 1 1 1 L(Aα , φ) := Dα φDα φ + Fαβ F αβ dxdt; 2 R4+1 2 4 here we are using the standard convention for raising indices. Introducing the covariant wave operator 2A := Dα Dα we can write the Maxwell-Klein Gordon system in the following form (1)
∂ β Fαβ = −Jα := =(φDα φ), 2A φ = 0
One key feature of this system is the underlying Gauge invariance, which is manifested by the fact that if (Aα , φ) is a solution, then so is (Aα − ∂α χ, eiχ φ). This allows us to impose an The first author was partially supported by the Swiss National Science Foundation. The second author was partially supported by the NSF grant DMS-1001675 . The third author was partially supported by the NSF grant DMS-0801261 and by the Simons Foundation. The first author would like to express his thanks to the University of California, Berkeley, for hosting him during the summer 2011 and 2012. 1The Lagrangian below should also contain a mass term, which we choose to neglect here; thus the system under consideration might be more aptly named “the Maxwell-scalar field system”. For historical reasons we have chosen to retain the “Maxwell-Klein Gordon” terminology. 1
additional Gauge condition, and we shall henceforth impose the Coulomb Gauge condition which requires (2)
4 X
∂j Aj = 0
j=1
The MKG-CG system can be written explicitly in the following form (3a)
2Ai = Pi Jx
(3b)
2A φ = 0
(3c)
∆A0 = J0
(3d)
∆∂t A0 = ∇i Ji ,
where the operator P denotes the Leray projection onto divergence free vector fields, P = I − ∇∆−1 ∇ The last equation (3d) is a consequence of (3c) due to the divergence free condition on the moments ∂ α Jα = 0, which in turn follows from (3b). However, it plays a role in the sequel so it is added here for convenience. Here it is assumed that the data Aj (0, ·) satisfy the vanishing divergence relation (2). We remark that given an arbitrary finite energy data set for the MKG problem, one can find a gauge equivalent data set of comparable size which satisfies the Coulomb gauge condition. This argument only involves solving linear elliptic pde’s, and is omitted. The key question now is to decide whether a family of data (A, φ)[0] := (Aα (0, ·), ∂t Aα (0, ·), φ(0, ·), ∂t φ(0, ·)) which satisfy the compatibility conditions required by the two equations for A0 above can be extended to a global-in-time solution for the Maxwell-Klein-Gordon system. The key for deciding this question is the criticality character of the system. Note that the energy Z 1X 2 1X E(A, φ) := Fαβ + |Dα φ|2 dx 2 α R4 4 α,β is preserved under the flow (1), and in our 4 + 1-dimensional setting it is also invariant under the natural scaling φ(t, x) → λφ(λt, λx), Aα (t, x) → λA(λt, λx) This means the 4+1-MKG system is energy critical, and in recent years a general approach to the large data Cauchy problem associated with energy critical wave equations has emerged. The first key step in this approach consists in establishing an essentially optimal global wellposedness result for data which are small in the energy norm, which is usually the optimal small-data global well-posedness result achievable. In this paper, we set out to prove this for the 4 + 1-dimensional Maxwell-Klein-Gordon system. Theorem 1. a) Let (A, φ)[0] be a C ∞ Coulomb data set satisfying (4)
E(A, φ) < ∗
for a sufficiently small universal constant ∗ > 0. Then the system (1) admits a unique global smooth solution (A, φ) on R4+1 with these data. 2
b) In addition, the data to solution operator extends continuously2 on the set (4) to a map H˙ 1 (R4 ) × L2 (R4 ) 3 (A, φ)[0] → (A, φ) ∈ C(R; H˙ 1 (R4 )) ∩ C˙ 1 (R, L2 (R4 )) We remark that the same result holds in all higher dimensions for small data in the scale n n invariant space H˙ 2 −1 × H˙ 2 −2 . This has already been known in dimensions n ≥ 6, see [21]. We have chosen to restrict our exposition to the more difficult case n = 4 in order to keep the notations simple, but our analysis easily carries over to dimension n = 5. On the other hand, we do not know whether a similar result holds in dimension n = 3. Before explaining some more details of our approach, we recall here earlier developments on this problem, and how our approach relates to these. Considering the case of general spatial dimension n and denoting the critical Sobolev exponent by sc = n2 − 1 (thus sc = 1 for n = 4, corresponding to the energy), a global regularity result for data which are smooth and small in H˙ sc was established in dimensions n ≥ 6 in the work [21] which served as inspiration to the present work. We note in passing that a result analogous to [21] was established in [18] for the Yang-Mills system in dimensions n ≥ 6, and the present work most likely also admits a corresponding analogue for the 4 + 1-dimensional Yang-Mills problem. The global regularity question for the physically relevant n = 3 case of the Yang-Mills problem had been established earlier in the groundbreaking work [6]. Observe that this problem is energy sub-critical. In the context of MKG, the result [21] had been preceded by a number of works which aimed at improving the local wellposedness of MKG in the n = 3 case, beginning with [9], followed by [5], and more recently [19]; the latter in particular established an essentially optimal local well-posedness result by exploiting a subtle cancellation feature of MKG, which also plays a role in the present work. We also mention the recent result [8] which establishes global regularity for energy sub-critical data in the n = 3 case, in the spirit of earlier work by Bourgain [3]. In the higher dimensional case n ≥ 4, an essentially optimal local well-posedness result for a model problem closely related to MKG was obtained in [14]. This model problem does not display the crucial cancellation feature of the precise MKG-system which enable us here to go all the way to the critical exponent and global regularity. We mention also that essentially optimal local well-posedness for the exact MKG-system was obtained in [17]. Finally, the recent work [24] established global regularity of equations of MKG-type with data small in a weighted but scaling invariant Besov type space, in the case n = 4. The present paper follows a similar strategy as [21] : one observes that the spatial Gauge connection components Aj , j = 1, 2, 3, which are governed by the first equation (3a), may in fact be decomposed into a free wave part and an inhomogeneous term (the second term in the Duhamel formula for A) which in fact obeys a better l1 -Besov type bound (while energy corresponds to l2 ) Aj = Afree + Anonlin j j This is important for handling the key difficulty of the MKG-system , which is the equation for φ, i. e. the second equation (3b). In fact, we shall verify that the contribution of the term Anonlin to the difficult magnetic interaction term 2iAj ∂j φ in the low-high frequency interaction case can be suitably bounded when combined with the term 2iA0 ∂t φ, an observation coming from [19]. However, the contribution of the free term Afree to the magnetic interaction term 2here
the continuity is locally in time 3
is nonperturbative and cannot be handled in this manner. Thus, following the example set in [21], we retain this term into the covariant wave operator3 2Afree . More precisely, we shall define a suitable paradifferential wave operator 2pA which incorporates the ’leading part’ of 2Afree while relegating the rest to the source terms on the right. The key novelty of this paper then is the development of a functional calculus, involving in particular X s,b -type as well as atomic null-frame spaces developed in other contexts, for solutions of the general inhomogeneous ’covariant’ wave equation 2pA = f This refined functional calculus is necessary to control the nonlinear interaction terms, which become significantly more delicate in the critical dimension than in the setting studied in [21]. In particular, the Strichartz norms themselves appear far from sufficient to handle the present situation. The above covariant wave equation will be solved by means of a suitable approximate parametrix, and we show that this parametrix satisfies many of the same bounds as the usual free wave parametrix, in particular encompassing refined square sum type microlocalized Strichartz norms as well as null-frame spaces. We expect the calculus developed here to be of fundamental importance in other contexts, such as the regularity question of the critical Yang-Mills system and related problems from mathematical physics. 2. Technical preliminaries Throughout the sequel we shall rely on Littlewood-Paley calculus, both in space as well as space-time. In particular, we constantly invoke the standard Littlewood-Paley localizers Pk , k ∈ Z, which are defined by |ξ| ˆ d P k f = χ( k )f (ξ) 2 4 for functions f defined on R . Here χ is a smooth bump function, supported on [ 14 , 4], which P satisfies the key condition k∈Z χ( 2ξk ) = 1 if ξ > 0. To measure proximity of the space-time Fourier support to the light cone, we use the concept of modulation. Thus we introduce the multipliers Qj , j ∈ Z, via |τ | − |ξ| d Q )fˆ(τ, ξ) j f (τ, ξ) = χ( 2j with the same χ as before, whereˆin this context denotes the space-time Fourier transform. We then refer to 2j as the modulation of the function. On occasion we shall also use multipliers Sl , which restrict the space-time frequency to size ∼ 2l . These multipliers allow us to introduce a variety of norms. In particular, for any p ∈ [1, ∞), we set for any norm k · kS X
1
F p :=
Pk F p p l S
S
k∈Z
We also have the following X s,b -type norms, applied to functions localized to spatial frequency ∼ 2k : X
p 1 p
F kX s,r := 2sk 2rj Qj F 2 , p ∈ [1, ∞), Lt,x
p
j∈Z 3Here
we set Afree = 0. 0 4
with the obvious analogue
F kX s,r := 2sk sup 2rj Qj F 2 ∞ L
t,x
k∈Z
For more refined norms, we shall also have to use multipliers Plω , which localize the homoξ to caps ω ⊂ S 3 of diameter 2l , by means of smooth cutoffs. In these geneous variable |ξ| situations, we shall assume that for each l a uniformly (in l) finitely overlapping covering of S 3 by caps ω has been chosen with appropriate cutoffs subordinate to these caps. Similar comments apply to the multipliers PC l00 which localize to rectangular boxes and will be dek fined below. Given a norm k · kS with corresponding space S, we denote by Sk the space of functions in S which are localized to frequency ∼ 2k . Furthermore, we denote by Sk,± the subspace of functions in Sk with Fourier support in the half-space τ >< 0, with τ the Fourier variable dual to t. 3. Function Spaces There are three function spaces we work with: N , N ∗ , and S. These are set up to that their dyadic subspaces Nk , Nk∗ , and Sk satisfy the following relations: 0,− 21
(5)
0, 12
Nk = L1 (L2 ) + X1
,
k F k2N =
X
X1
⊆ Sk ⊆ Nk∗ ,
Then define: k Pk F k2Nk .
k
We also define
Sk]
by kukS ] = k2ukNk + k∇ukL∞ L2 k
On occasion we need to separate the two characteristic cones {τ = ±|ξ|}. Thus we define Nk = Nk,+ ∩ Nk,−
Nk,± , ] Sk,± , ∗ Nk,± ,
] ] Sk] = Sk,+ + Sk,− ∗ ∗ Nk∗ = Nk,+ + Nk,−
Our space Sk scales like L2 free waves, and is defined by: k φ k2Sk = k φ k2S str + k φ k2S ang + k φ k2 0, 12 , k
k
X∞
where: (6)
1
4
Skstr = ∩ 1 + 3/2 6 3 2( q + r −2)k Lq (Lr ) , q
r
k φ k2S ang = sup k
4
X
l
